scholarly journals Generalized Characteristic Polynomials of Join Graphs and Their Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Pengli Lu ◽  
Ke Gao ◽  
Yang Yang
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Characteristic Polynomials ◽  
Electrical Networks ◽  
Kirchhoff Index ◽  
Laplacian Spectra ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Number Of Spanning Trees

The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices ofGin electrical networks.LEL(G)is the Laplacian-Energy-Like Invariant ofGin chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex joinG1⊚G2and the subdivision-edge-edge joinG1⊝G2. We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials ofG1⊚G2andG1⊝G2whenG1isr1-regular graph andG2isr2-regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, andLELofG1⊚G2andG1⊝G2in terms of the Laplacian spectra ofG1andG2.

scholarly journals On the spectra of a new duplication based corona of graphs

2021 ◽  
Vol 27 (1) ◽  
pp. 208-220
Author(s):  
Renny P. Varghese ◽  
◽  
D. Susha ◽  
Keyword(s):  
Spanning Trees ◽  
Graph Product ◽  
Kirchhoff Index ◽  
Cospectral Graphs ◽  
Laplacian Spectra ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Product Of Graphs ◽  
Number Of Spanning Trees

In this paper we introduce a new corona-type product of graphs namely duplication corresponding corona. Here we mainly determine the adjacency, Laplacian and signless Laplacian spectra of the new graph product. In addition to that, we find out the incidence energy, the number of spanning trees, Kirchhoff index and Laplacian-energy-like invariant of the new graph. Also we discuss some new classes of cospectral graphs.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950056
Author(s):  
Shreekant Patil ◽  
Mallikarjun Mathapati
Keyword(s):  
Spanning Trees ◽  
Product Formula ◽  
Regular Graphs ◽  
Kirchhoff Index ◽  
Cospectral Graphs ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Graph Operation ◽  
Product Of Graphs ◽  
Number Of Spanning Trees

Recently Indulal and Balakrishnan [Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13 (2016) 230–234] put forward a new graph operation, namely, the Indu–Bala product [Formula: see text] of graphs [Formula: see text] and [Formula: see text], and it is obtained from two disjoint copies of the join [Formula: see text] of [Formula: see text] and [Formula: see text] by joining the corresponding vertices in the two copies of [Formula: see text]. In this paper, we obtain the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of [Formula: see text] in terms of the corresponding spectra of [Formula: see text] and [Formula: see text]. As applications, these results enable us to construct infinitely many pairs of respective cospectral graphs. Further, the Laplacian spectra enable us to get the formulas of the number of spanning trees and Kirchhoff index of [Formula: see text] in terms of the Laplacian spectra of regular graphs [Formula: see text] and [Formula: see text].

scholarly journals Some Chemistry Indices of Clique-Inserted Graph of a Strongly Regular Graph

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Chun-Li Kan ◽  
Ying-Ying Tan ◽  
Jia-Bao Liu ◽  
Bao-Hua Xing
Keyword(s):  
Regular Graph ◽  
Spanning Trees ◽  
Strongly Regular Graph ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Triangular Graph ◽  
Signless Laplacian ◽  
Strongly Regular ◽  
Number Of Spanning Trees

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

Signless Laplacian spectral characterization of some joins

2015 ◽  
Vol 30 ◽  
Author(s):  
Xiaogang Liu ◽  
Pengli Lu
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spectral Characterization ◽  
Laplacian Eigenvalue ◽  
Laplacian Spectra ◽  
Signless Laplacian

The join of two disjoint graphs G and H, denoted by G ∨ H, is the graph obtained by joining each vertex of G to each vertex of H. In this paper, the signless Laplacian characteristic polynomial of the join of two graphs is first formulated. And then, a lower bound for the i-th largest signless Laplacian eigenvalue of a graph is given. Finally, it is proved that G ∨ K_m, where G is an (n − 2)-regular graph on n vertices, and K_n ∨ K_2 except for n = 3, are determined by their signless Laplacian spectra.

Study on Applications of Laplacian Spectra for a Network

2013 ◽  
Vol 753-755 ◽  
pp. 2859-2862
Author(s):  
Hai Tang Wang
Keyword(s):  
Recurrence Relations ◽  
Spanning Trees ◽  
Small World ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Scale Free ◽  
Structure And Dynamics ◽  
Laplacian Spectra ◽  
Number Of Spanning Trees ◽  
Biological Field

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.

scholarly journals Spectra of Subdivision Vertex-Edge Join of Three Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 171 ◽  
Author(s):  
Fei Wen ◽  
You Zhang ◽  
Muchun Li
Keyword(s):  
Spanning Trees ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Graph Operation ◽  
Arbitrary Graphs ◽  
Number Of Spanning Trees

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

scholarly journals The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-10
Author(s):  
Jia-Bao Liu ◽  
Jing Chen ◽  
Jing Zhao ◽  
Shaohui Wang
Keyword(s):  
Spanning Trees ◽  
Characteristic Polynomials ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Number Of Spanning Trees ◽  
Structure Properties

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Spectra of the generalized edge corona of graphs

2018 ◽  
Vol 10 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Yanyan Luo ◽  
Weigen Yan
Keyword(s):  
Characteristic Polynomial ◽  
Spanning Trees ◽  
Simple Graph ◽  
Simple Graphs ◽  
Signless Laplacian ◽  
Number Of Spanning Trees

Let [Formula: see text] be a simple graph with [Formula: see text] edges and [Formula: see text], [Formula: see text] be [Formula: see text] simple graphs. The generalized edge corona, denoted by [Formula: see text], is the graph obtained by taking one copy of graphs [Formula: see text], [Formula: see text], [Formula: see text] and then joining two end-vertices of the [Formula: see text]th edge [Formula: see text] of [Formula: see text] to every vertex of [Formula: see text] for [Formula: see text]. In this paper, we determine and study the characteristic polynomial, Laplacian polynomial and signless Laplacian polynomial of [Formula: see text]. As an application, we also count the number of spanning trees of the generalized edge corona.

scholarly journals Properties of Commuting Graphs over Semidihedral Groups

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 103
Author(s):  
Tao Cheng ◽  
Matthias Dehmer ◽  
Frank Emmert-Streib ◽  
Yongtao Li ◽  
Weijun Liu
Keyword(s):  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Vertex Connectivity ◽  
Laplacian Energy ◽  
Number Of Spanning Trees

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

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